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In a given solubility equilibrium, I know that if $ Q >K_s$, then the salt will precipitate until $ Q =K_s$. If $ Q <K_s$, however, would the precipitate dissolve until $ Q =K_s$ too?

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You've described the general tendency for any equilibrium process, including solubility. So, the short answer is yes, you will dissolve more until the reaction quotient is equal to the equilibrium constant.

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